arXiv subjects
Publications and source records attributed to Rodney G. Downey.
The Ziegler degrees were introduced to characterize definability in group theory. In [JLSta], the authors show that the first order-theory of the Ziegler degrees (as a partial order) is undecidable. We improve this result, showing that the theory of the Ziegler degrees is bi-interpretable with true second-order arithmetic.
Cousin's lemma is a compactness principle that naturally arises when studying the gauge integral, a generalisation of the Lebesgue integral. We study the axiomatic strength of Cousin's lemma for various classes of functions, using Friedman and Simpson's reverse mathematics in second-order arithmetic. We prove that, over $\mathsf{RCA}_0$: (i) Cousin's lemma for continuous functions is equivalent to $\mathsf{WKL}_0$; (ii) Cousin's lemma for Baire class 1 functions is equivalent to $\mathsf{ACA}_0$; (iii) Cousin's lemma for Baire class 2 functions, or for Borel functions, are both equivalent to $\mathsf{ATR}_0$ (modulo some induction).
Kobayashi introduced a uniform notion of compressibility of infinite binary sequences in terms of relative Turing computations with sub-identity use of the oracle. Kobayashi compressibility has remained a relatively obscure notion, with the exception of some work on resource bounded Kolmogorov complexity. The main goal of this note is to show that it is relevant to a number of topics in current research on algorithmic randomness. We prove that Kobayashi compressibility can be used in order to define Martin-Loef randomness, a strong version of finite randomness and Kurtz randomness, strictly in terms of Turing reductions. Moreover these randomness notions naturally correspond to Turing reducibility, weak truth-table reducibility and truth-table reducibility respectively. Finally we discuss Kobayashi's main result from his 1981 technical report regarding the compressibility of computably enumerable sets, and provide additional related original results.
This work contributes to the programme of studying effective versions of "almost everywhere" theorems in analysis and ergodic theory via algorithmic randomness. We determine the level of randomness needed for a point in a Cantor space $ \{0,1\}^{\NN}$ with the uniform measure and the usual shift so that effective versions of the multiple recurrence theorem of Furstenberg holds for iterations starting at the point. We consider recurrence into closed sets that possess various degrees of effectiveness: clopen, $\PPI$ with computable measure, and $\PPI$. The notions of Kurtz, Schnorr, and \ML\ randomness, respectively, turn out to be sufficient. We obtain similar results for multiple recurrence with respect to the $k$ commuting shift operators on $\{0,1\}^{\NN^{\normalsize k}}$.
We give an analog of the Myhill-Nerode methods from formal language theory for hypergraphs and use it to derive the following results for two NP-hard hypergraph problems: * We provide an algorithm for testing whether a hypergraph has cutwidth at most k that runs in linear time for constant k. In terms of parameterized complexity theory, the problem is fixed-parameter linear parameterized by k. * We show that it is not expressible in monadic second-order logic whether a hypergraph has bounded (fractional, generalized) hypertree width. The proof leads us to conjecture that, in terms of parameterized complexity theory, these problems are W[1]-hard parameterized by the incidence treewidth (the treewidth of the incidence graph). Thus, in the form of the Myhill-Nerode theorem for hypergraphs, we obtain a method to derive linear-time algorithms and to obtain indicators for intractability for hypergraph problems parameterized by incidence treewidth. In an appendix, we point out an error and a fix to the proof of the Myhill-Nerode theorem for graphs in Downey and Fellow's book on parameterized complexity.
In graph theory, Courcelle's theorem essentially states that, if an algorithmic problem can be formulated in monadic second-order logic, then it can be solved in linear time for graphs of bounded treewidth. We prove such a metatheorem for a general class of triangulations of arbitrary fixed dimension d, including all triangulated d-manifolds: if an algorithmic problem can be expressed in monadic second-order logic, then it can be solved in linear time for triangulations whose dual graphs have bounded treewidth. We apply our results to 3-manifold topology, a setting with many difficult computational problems but very few parameterised complexity results, and where treewidth has practical relevance as a parameter. Using our metatheorem, we recover and generalise earlier fixed-parameter tractability results on taut angle structures and discrete Morse theory respectively, and prove a new fixed-parameter tractability result for computing the powerful but complex Turaev-Viro invariants on 3-manifolds.
We study connections between classical asymptotic density and c.e. sets. We prove that a c.e. Turing degree d is not low if and only if d contains a c.e. set A of density 1 which has no computable subsets of density 1, giving a natural characterization of non-low c.e. degrees. In contrast, we prove that every nonzero c.e. degree contains a set which is generically computable but not coarsely computable. There is a very close connection between the computational complexity of a set and the computational complexity of its density as a real number where we measure complexity of real numbers as the position of their left Dedekind cuts in the Arithmetic Hierarchy. We characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study "computable at density r" where r is an arbitrary real number in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity and cohesiveness.
One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that some parameters contribute more intractability than others and it seems reasonable to develop a theory of computational complexity which seeks to exploit this fact. Such a theory should be able to address the needs of practicioners in algorithmics. The last twenty years have seen the development of such a theory. This theory has a large number of successes in terms of a rich collection of algorithmic techniques both practical and theoretical, and a fine-grained intractability theory. Whilst the theory has been widely used in a number of areas of applications including computational biology, linguistics, VLSI design, learning theory and many others, knowledge of the area is highly varied. We hope that this article will show both the basic theory and point at the wide array of techniques available. Naturally the treatment is condensed, and the reader who wants more should go to the texts, Downey and Fellows, Flum and Grohe, Niedermeier, and the upcoming undergraduate text Downey and Fellows.
There are noncomputable c.e.\ sets, computable from every SJT-hard c.e.\ set. This yields a natural pseudo-jump operator, increasing on all sets, which cannot be inverted back to a minimal pair or even avoiding an upper cone.